Enumeration of Latin arrays. I: Case \(n\leq 3\) (Q1176064)

The authors introduce Latin arrays as a generalization of Latin squares. Let \(A\) be an \(n\times nk\) matrix defined on \(N=\{1,2,\ldots,n\}\). If each element of \(N\) occurs exactly once in each column of \(A\) and \(k\) times in each row of \(A\), then \(A\) is said to be an \((n,k)\) Latin array. This paper deals with the problem of enumeration of Latin arrays. A possible field of application of the results in this paper is cryptography. For Part II, see [Sci. China, Ser. A 34, No. 1, 20--29 (1991; Zbl 0738.05023)].